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forces, which makes it relatively easy to convert the force equations to
explicit ODEs (this has been demonstrated in section 3.4 for the Beaver model),
provided the translational equations are written in terms of a and b instead of
v and w.
2. A higher accuracy of the numerical computations can be achieved by relating
the translational motions to the flight-path axes. For agile aircraft having an upper
limit of the pitch rate q of about 2 rad s−1, and flying at high airspeeds (e.g.
V = 600 ms−1), the term u q in equation (2.28) may become as large as 120 g!
On the other hand, the factor Fz/m, which represents the normal acceleration
due to the external force along the ZB-axis (primarily gravity and aerodynamic
lift) has an upper-limit of only a few g’s. Hence, ‘artificial’ accelerations of
much greater magnitude than the actual physical accelerations of the aircraft
are introduced in the equations for u, v, and w, because of the high rotation
rates of the body-axes. In practice this means less favourable computer scaling
and hence poorer accuracy for a given computer precision if the simulation
model is based upon body-axes velocity components [16].
Because of said advantages, we will later treat V, a, and b as state variables in the
resulting nonlinear state-space model of the aircraft, while u, v, and w (and their
time-derivatives) will be treated as output variables. The first three variables are required
to solve the equations of motion; the other ones provide useful additional
information, that allows us to determine e.g. airplane drift due to wind and atmospheric
turbulence.
2.2. Expressing translational motions in flight-path axes 19
V
-X
D
L
-Za
a
a
Figure 2.2: Relationship between the aerodynamic forces in flight-path and body-axes
2.2.2 Expressing forces and velocities in terms of flight-path axes
Transforming the forces and velocities from body to flight-path axes is quite straightforward
(see ref.[14] for a detailed description). From figure 2.1, we can observe that
the body-axes velocity components are equal to:
2
4
uvw
3
5 = V
2
4
cos a cos b
sin b
sin a cos b
3
5 (2.31)
Hence:
V =
p
u2 + v2 + w2 (2.32)
a = arctan
w
u

(2.33)
b = arctan

v
p
u2 + w2

(2.34)
20 Chapter 2. Rigid body equations of motion
Aerodynamic forces and moments are commonly expressed in terms of aerodynamic
lift L, drag D, and sideforce Y, which are aligned along the flight-path axes ZW (in
negative direction, i.e. upwards), XW, and YW, respectively. However, for simulation
purposes, it is more convenient to use the body-axes force-components Xa, Ya, and
Za instead. The relationship between these variables has been shown in figure 2.2.
The body-axes components can be derived from the flight-path axes components by
means of the following axis-transformation:
2
4
Xa
Ya
−Za
3
5 =
2
4
−cos a 0 sin a
0 1 0
sin a 0 cos a
3
5 ·
2
4
D
Y
L
3
5 (2.35)
Notice the minus sign for the aerodynamic force component along the ZB-axis, which
is due to the fact that the positive ZB-axis points downwards.
2.2.3 Derivation of the ˙V -equation
From equation (2.32) we can deduce that:
˙V
= uu˙ + vv˙ + ww˙
V
(2.36)
Substituting the definitions (2.31) for u, v, and w, and cancelling terms yields:
˙V
= u˙ cos a cos b + v˙ sin b + w˙ sin a cos b (2.37)
If we substitute equations (2.28) for u˙, v˙, and w˙ , the terms involving the vehicle rotational
rates p, q, and r turn out to be identically zero, which becomes obvious after
again substituting equation (2.31) for u, v, and w. The resulting equation becomes:
˙V
= 1
m
􀀀
Fx cos a cos b + Fy sin b + Fz sin a cos b


(2.38)
2.2.4 Derivation of the ˙a-equation
Differentiating equation (2.33) with respect to the time yields:
˙a
= uw˙ − u˙w
u2 + w2 (2.39)
Using equation (2.31) we can re-write the denominator of this equation:
u2 + w2 = V2 − v2 = V2(1 − sin2 b) = V2 cos2 b (2.40)
Substituting the u and w-relations from equation (2.31) and equation (2.40) into equation
(2.39) yields:
˙a
=
w˙ cos a − u˙ sin a
V cos b
(2.41)
Substituting equations (2.28) for u˙ and w˙ , and rewriting terms yields:
˙a
= 1
V cos b

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